Vaidya walk: A sampling algorithm based on the volumetric barrier

The problem of sampling from the uniform distribution over a polytope arises in various contexts. We propose a new random walk for this purpose, which we refer to as the Vaidya walk, since it is based on the volumetric-logarithmic barrier introduced by Vaidya in the context of interior point methods for optimization. We show that the Vaidya walk mixes in significantly fewer steps compared to the Dikin walk, a random walk previously studied by Kannan and Narayanan. In particular, we prove that for a polytope in R
^d
defined by n constraints, the Vaidya walk mixes in O (√n/d) fewer steps than the Dikin walk. The per iteration cost for our method is at most twice that of the Dikin walk, and hence the speed up is significant for polytopes with n ≫ d. Furthermore, the algorithm is also faster than the Ball walk and Hit-and-Run for a large family of polytopes. We illustrate the speed-up of the Vaidya walk over the Dikin walk via several numerical examples and discuss possible new and faster algorithms for sampling from polytopes.